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ideal in a monoid

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Definition

Given a monoid (or semigroup) SS, a left ideal in SS is subsetAA of SS such that SAS A is contained in AA. Similarly, a right ideal is a subset AA such that AS⊆AA S \subseteq A. Finally, a two-sided ideal, or simply ideal, in SS is a subset AA that is both a left ideal and a right ideal.

Given a monoidal category(C,⊗,I)(C, \otimes, I) and a monoid object (or semigroup object) SS of CC, we can internalise the above. For instance, if m:S⊗S→Sm: S \otimes S \to S is the binary multiplication and μ=m∘(m⊗1S)=m∘(1S⊗m):S⊗S⊗S→S\mu = m \circ (m \otimes 1_S) = m \circ (1_S \otimes m): S \otimes S \otimes S \to S the ternary multiplication, a two-sided ideal is a subobjectAA of SS, i.e., a mono i:A→Si: A \to S in CC, such that the composite

factors through i:A→Si: A \to S. Clearly i:A→Si: A \to S is not necessarily a submonoid, inasmuch as the monoid unit e:I→Se: I \to S need not factor through i:A→Si: A \to S.

In particular for C=C = Ab, a monoid in CC is a ring and the corresponding notion of ideal in a ring is the most common notion of ideal.

See ideal for ideals in more well known contexts: commutative idempotent monoids (semilattices) and monoids in Ab (rings).

Properties and constructions

An ideal AA (on either side) must be a subsemigroup? of SS, but it is a submonoid iff 1∈A1 \in A, in which case A=SA = S.

Ideals forming a quantale

(Two-sided) ideals of a monoid AA are frequently the elements of a quantale whose multiplication is called taking the product of ideals. In the classical case of ideals over a ringRR, the product IJI J of ideals I,J⊆RI, J \subseteq R is the smallest ideal containing all products ij:i∈I,j∈Ji j: i \in I, j \in J; the sup-lattice of such ideals ordered by inclusion is a residuated lattice?, in that there are also division operations where

Let C\mathbf{C} be a well-poweredregularcosmos (‘cosmos’ in the sense of complete cocomplete symmetric monoidal closed category). Just using the fact that C\mathbf{C} is a cosmos, we may construct a monoidal bicategoryMod(C)Mod(\mathbf{C}) whose objects are monoids SS in C\mathbf{C}, whose 1-morphisms S→TS \to T are left-SS right-TTmodules, and whose 2-morphisms are bimodule homomorphisms.

For each monoid SS, there is a subbicategory of Mod(C)Mod(\mathbf{C}) whose only object is SS; this is a complete and cocomplete biclosed monoidal categoryModSMod_S whose objects are bimodules, i.e., 1-morphisms S→SS \to S in Mod(C)Mod(\mathbf{C}), and whose morphisms are bimodule homomorphisms. The unit of the monoidal product is SS with its standard SS-bimodule structure, and hence the sliceModS/SMod_S/S (see also semicartesian monoidal category) forms another complete and cocomplete biclosed monoidal category.

An ideal of SS is just a subobject of SS in ModSMod_S. Under the assumption that C\mathbf{C} is well-powered, the category of subobjects Sub(S)↪ModS/SSub(S) \hookrightarrow Mod_S/S is a (small) sup-lattice. Under the regularity assumption on C\mathbf{C}, the subcategory Sub(S)↪ModS/SSub(S) \hookrightarrow Mod_S/S is reflective, and by applying the reflector to the monoidal product on ModS/SMod_S/S, we obtain a product on Sub(S)Sub(S) which preserves arbitrary joins in each variable, hence a quantale. The unit of the quantale is the top element, namely SS considered as an ideal.